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<li><a href="#%E4%BF%A1%E5%8F%B7%E4%B8%8E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F">&#x4FE1;&#x53F7;&#x4E0E;&#x7EBF;&#x6027;&#x7CFB;&#x7EDF;</a>
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#x5C;mathit{F}$</span></a></li>
<li><a href="#%E4%BB%8E%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E5%88%B0%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2mathitftomathscrftoc">&#x4ECE;&#x5085;&#x91CC;&#x53F6;&#x7EA7;&#x6570;&#x5230;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;<span class="mathjax-exps"><p>[MUMETOC]</p>
#x5C;mathit{F}\to\mathscr{F}$</span></a>
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<li><a href="#%E5%82%85%E9%87%8C%E5%8F%B6%E7%B3%BB%E6%95%B0%E5%92%8C%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E5%85%B3%E7%B3%BBtoc">&#x5085;&#x91CC;&#x53F6;&#x7CFB;&#x6570;&#x548C;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;&#x7684;&#x5173;&#x7CFB;</a></li>
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#x5C;mathscr{F}\to\mathscr{L}$</span></a></li>
<li><a href="#%E4%BB%8E%E6%8B%89%E6%B0%8F%E5%8F%98%E6%8D%A2%E5%88%B0z%E5%8F%98%E6%8D%A2mathscrltomathscrztoc">&#x4ECE;&#x62C9;&#x6C0F;&#x53D8;&#x6362;&#x5230;z&#x53D8;&#x6362;<span class="mathjax-exps"><p>[MUMETOC]</p>
#x5C;mathscr{L}\to\mathscr{z}$</span></a></li>
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<p><span id="toc"></span></p>
<h1 class="mume-header" id="%E4%BF%A1%E5%8F%B7%E4%B8%8E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F">&#x4FE1;&#x53F7;&#x4E0E;&#x7EBF;&#x6027;&#x7CFB;&#x7EDF;</h1>

<p><span class="mathjax-exps">$&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6; \longrightarrow \mathit{F} \longrightarrow \mathscr{F} \longrightarrow \mathscr{L} \longrightarrow \mathscr{z}$</span></p>
<h2 class="mume-header" id="%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0%E9%9B%86toc"><a href="#toc">&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;</a></h2>

<h3 class="mume-header" id="%E4%BB%80%E4%B9%88%E6%98%AF%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0toc"><a href="#toc">&#x4EC0;&#x4E48;&#x662F;&#x6B63;&#x4EA4;&#x51FD;&#x6570;</a></h3>

<p>&#x8BB0;&#x5B9A;&#x4E49;&#x5728;&#x533A;&#x95F4;<span class="mathjax-exps">$[t_1,t_2]$</span>&#x4E0A;&#x7684;&#x4E24;&#x4E2A;&#x51FD;&#x6570;<span class="mathjax-exps">$\psi_1$</span>&#x548C;<span class="mathjax-exps">$\psi_2$</span>, &#x82E5;&#x6EE1;&#x8DB3;<span class="mathjax-exps">$\int_{t1}^{t2} \psi_1 \cdot \psi_2^{\ast}=0$</span>, &#x5219;&#x8FD9;&#x4E24;&#x4E2A;&#x51FD;&#x6570;&#x662F;&#x6B63;&#x4EA4;&#x51FD;&#x6570;</p>
<h3 class="mume-header" id="%E4%BB%80%E4%B9%88%E6%98%AF%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0%E9%9B%86toc"><a href="#toc">&#x4EC0;&#x4E48;&#x662F;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;</a></h3>

<p>&#x6709;<span class="mathjax-exps">$n$</span>&#x4E2A;&#x51FD;&#x6570;<span class="mathjax-exps">$\{\psi_1,\psi_2,\cdots,\psi_n\}$</span>&#x6784;&#x6210;&#x4E00;&#x4E2A;&#x51FD;&#x6570;&#x96C6;, &#x8BB0;&#x4E3A;<span class="mathjax-exps">$\Psi$</span>. &#x5F53;&#x8FD9;<span class="mathjax-exps">$n$</span> &#x4E2A;&#x51FD;&#x6570;&#x5728;&#x5B9A;&#x4E49;&#x57DF;<span class="mathjax-exps">$[t_1, t_2]$</span>&#x5185;&#x6EE1;&#x8DB3;&#x5982;&#x4E0B;&#x5173;&#x7CFB;, &#x5219;&#x79F0;<span class="mathjax-exps">$\Psi$</span>&#x4E3A;<span class="mathjax-exps">$[t_1,t_2]$</span>&#x5185;&#x7684;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;. &#x5982;&#x679C;&#x5728;<span class="mathjax-exps">$\Psi$</span>&#x4E4B;&#x5916;&#x627E;&#x5230;&#x51FD;&#x6570;<span class="mathjax-exps">$\phi$</span>, &#x4F7F;&#x5F97;<span class="mathjax-exps">$\int_{t_1}^{t_2} \phi \cdot \psi_i^{\ast}=0$</span>, &#x90A3;&#x4E48;<span class="mathjax-exps">$\Psi$</span>&#x4E3A;&#x5B8C;&#x5907;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;.</p>
<p></p><div class="mathjax-exps">$$\int_{t_1}^{t_2} \psi_i \cdot \psi_j^{\ast} = \begin{cases} 0, when \ i \neq j \\  k_i \neq 0, when \  i = j \end{cases}$$</div><p></p>
<p><strong>&#x5173;&#x4E8E;&#x5B8C;&#x5907;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;&#x7684;&#x4E00;&#x4E2A;&#x91CD;&#x8981;&#x5B9A;&#x7406;</strong></p>
<p><strong>&#x5B9A;&#x7406;</strong>: &#x8BB0;&#x6709;&#x4E00;&#x4E2A;&#x7531;<span class="mathjax-exps">$n$</span>&#x4E2A;&#x51FD;&#x6570;&#x7EC4;&#x6210;&#x7684;&#x5B8C;&#x5907;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;<span class="mathjax-exps">$\{\psi_i\}(i=1,2,\cdots,n)$</span>, &#x90A3;&#x4E48;&#x5BF9;&#x4E8E;&#x4EFB;&#x4E00;&#x7EDD;&#x5BF9;&#x53EF;&#x79EF;&#x7684;&#x51FD;&#x6570;<span class="mathjax-exps">$f(t)$</span>, &#x53EF;&#x901A;&#x8FC7;&#x8FD9;<span class="mathjax-exps">$n$</span>&#x4E2A;&#x51FD;&#x6570;&#x7684;&#x7EBF;&#x6027;&#x7EC4;&#x5408;&#x6765;&#x8868;&#x793A;. &#x5373;&#x6709;:</p>
<p></p><div class="mathjax-exps">$$f(t) = \lim_{n \to \infty} \sum_{i=1}^{n} C_i \cdot \psi_i, C_i \in \mathbb{R}$$</div><p></p>
<p>&#x5047;&#x8BBE;<span class="mathjax-exps">$f(t)=C_1 \psi_1 + C_2 \psi_2 + \cdots$</span>, &#x4E3A;&#x5747;&#x65B9;&#x8BEF;&#x5DEE;<span class="mathjax-exps">$\epsilon^{2}=\frac{1}{t_2 - t_1} \int_{t_1}^{t_2}(f(t) - \sum_{i=1}^{\infty} C_i \cdot \psi_i)^2 \rm dt$</span>&#x6700;&#x5C0F;, &#x6709;:</p>
<p></p><div class="mathjax-exps">$$\begin{matrix} \frac{\partial{\epsilon^2}}{\partial{C_j}} = 0 \Rightarrow \\ -2\int_{t_1}^{t_2} (f(t) - C_j \psi_j) \psi_j \rm dt= -2\int_{t_1}^{t_2}f(t) \psi_j \rm dt + 2C_j\int_{t_1}^{t_2}\psi_j^2 = 0 \Rightarrow \\ C_j = \frac{\int_{t_1}^{t_2}f(t)\psi_j \rm dt}{\int_{t_1}^{t_2}\psi_j^2 \rm dt} \end{matrix}$$</div><p></p>
<p>&#x53E6;&#x6839;&#x636E;&#x5747;&#x65B9;&#x8BEF;&#x5DEE;&#x65B9;&#x7A0B;, &#x6709;: <span class="mathjax-exps">$\int_{t_1}^{t_2}f^2(t) \rm dt = \sum_{i=1}^{\infty}|\int_{t_1}^{t_2}|C_i \psi_i|^2 \rm dt$</span>.</p>
<h3 class="mume-header" id="%E5%B8%B8%E8%A7%81%E5%AE%8C%E5%A4%87%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0%E9%9B%86toc"><a href="#toc">&#x5E38;&#x89C1;&#x5B8C;&#x5907;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;</a></h3>

<ul>
<li>&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x96C6;:</li>
</ul>
<p></p><div class="mathjax-exps">$$\{ 1, \cos{\Omega t}, \cdots, \cos{m \Omega t}, \cdots, \sin{\Omega t}, \cdots, \sin{n \Omega t}, \cdots \}, t \in [t_0, t_0 + T], T = \frac{2 \pi}{\Omega}$$</div><p></p>
<ul>
<li>&#x6307;&#x6570;&#x51FD;&#x6570;&#x96C6;:</li>
</ul>
<p></p><div class="mathjax-exps">$${e^{jn \Omega t}}, t \in {t_0, t_0 + T}, T = \frac{2 \pi}{\Omega}, n=0,\pm 1, \pm 2, \cdots$$</div><p></p>
<ul>
<li>&#x5176;&#x5B83;&#x7684;&#x8BF8;&#x5982;Legendre&#x591A;&#x9879;&#x5F0F;, Hermite&#x591A;&#x9879;&#x5F0F;, Laguerrel&#x591A;&#x9879;&#x5F0F;&#x7B49;&#x592A;&#x590D;&#x6742;&#x4E86;, &#x5F88;&#x5C11;&#x7528;&#x5230;, &#x6545;&#x4E0D;&#x5217;;</li>
</ul>
<h2 class="mume-header" id="%E4%BB%8E%E6%AD%A3%E4%BA%A4%E5%87%BD%E6%95%B0%E9%9B%86%E5%88%B0%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0mathitftoc"><a href="#toc">&#x4ECE;&#x6B63;&#x4EA4;&#x51FD;&#x6570;&#x96C6;&#x5230;&#x5085;&#x91CC;&#x53F6;&#x7EA7;&#x6570;<span class="mathjax-exps">$\mathit{F}$</span></a></h2>

<p>&#x5047;&#x8BBE;&#x6709;&#x4E00;&#x5468;&#x671F;&#x51FD;&#x6570;<span class="mathjax-exps">$f(t)$</span>, &#x5176;&#x5468;&#x671F;&#x4E3A;<span class="mathjax-exps">$T, \Omega = \frac{2\pi}{T}$</span>, &#x5176;&#x5728;<span class="mathjax-exps">$[-\frac{T}{2}, \frac{T}{2}]$</span>&#x7EDD;&#x5BF9;&#x53EF;&#x79EF;.  &#x4F7F;&#x7528;&#x5B8C;&#x5907;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x96C6;&#x5BF9;&#x5176;&#x8FDB;&#x884C;&#x6B63;&#x4EA4;&#x5C55;&#x5F00;, &#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$f(t) = c_0 + \sum_{n=1}^{\infty}a_n \cos{n\Omega t} + \sum_{n=1}^{\infty}b_n \sin{n\Omega t}$$</div><p></p>
<p>&#x5C06;<span class="mathjax-exps">$f(t), \psi_i$</span>&#x5E26;&#x5165;<span class="mathjax-exps">$C_j$</span>, &#x6C42;&#x5F97;&#x7CFB;&#x6570;:<br>
</p><div class="mathjax-exps">$$\begin{cases} a_n = \frac{2}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t) \cos{n\Omega t} \rm dt \\ b_n = \frac{2}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t) \sin{n\Omega t} \rm dt \\ c_0 = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t) \rm dt = \frac{a_0}{2} \end{cases}$$</div><p></p>
<p>&#x4EE5;&#x4E0A;&#x4FBF;&#x662F;&#x5085;&#x91CC;&#x53F6;&#x7EA7;&#x6570;&#x7684;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x5F62;&#x5F0F;, <span class="mathjax-exps">$a_n, b_n$</span>&#x79F0;&#x4E3A;&#x5085;&#x91CC;&#x53F6;&#x7CFB;&#x6570;.</p>
<p>&#x53C8;&#x53EF;&#x901A;&#x8FC7;&#x6B27;&#x62C9;&#x516C;&#x5F0F;, &#x5C06;&#x4E09;&#x89D2;&#x51FD;&#x6570;&#x5F62;&#x5F0F;&#x5316;&#x4E3A;&#x6307;&#x6570;&#x5F62;&#x5F0F;, &#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{matrix} f(t) = \frac{A_0}{2} + \sum_{n=1}^{\infty}A_n \cos{(n\Omega t + \theta_n)} \Rightarrow \\ f(t) = \frac{A_0}{2} + \frac{A_n}{2}\sum_{n=1}^{\infty}(e^{jn\Omega t}e^{j\theta_n}+e^{-jn\Omega t}e^{-j\theta_n}) \Rightarrow \\ f(t) = \sum_{n=-\infty}^{\infty}\frac{1}{2}A_n e^{j\theta_n} e^{jn\Omega t}  \end{matrix}$$</div><p></p>
<p>&#x4EE5;&#x4E0A;&#x4FBF;&#x662F;&#x5085;&#x91CC;&#x53F6;&#x7EA7;&#x6570;&#x7684;&#x6307;&#x6570;&#x5F62;&#x5F0F;, &#x5176;&#x4E2D;<span class="mathjax-exps">$F_n$</span>&#x79F0;&#x4E3A;&#x5085;&#x91CC;&#x53F6;&#x7CFB;&#x6570;, &#x8FDB;&#x4E00;&#x6B65;&#x5316;&#x7B80;&#x6709;&#x5982;&#x4E0B;&#x5F62;&#x5F0F;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} F_n&amp; =\frac{1}{2}(A_n \cos{\theta_n} + jA_n \sin{\theta_n}) \\ &amp; = \frac{1}{2}(a_n - jb_n) \\ &amp; = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}(f(t)\cos{n\Omega t} - jf(t)\sin{n\Omega t}) \mathrm{dt}  \\ &amp; = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t)e^{-jn\Omega t} \rm{dt} \end{aligned}$$</div><p></p>
<p>&#x7EFC;&#x4E0A;: &#x6307;&#x6570;&#x5F62;&#x5F0F;&#x7684;&#x5085;&#x91CC;&#x53F6;&#x7EA7;&#x6570;&#x6B63;&#x9006;&#x53D8;&#x6362;&#x6574;&#x7406;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{cases} f(t)=\sum_{n=-\infty}^{\infty}F_n e^{jn\Omega t} \\ F_n = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f(t)e^{-jn\Omega t} \rm{dt} \end{cases}$$</div><p></p>
<h2 class="mume-header" id="%E4%BB%8E%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0%E5%88%B0%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2mathitftomathscrftoc"><a href="#toc">&#x4ECE;&#x5085;&#x91CC;&#x53F6;&#x7EA7;&#x6570;&#x5230;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;<span class="mathjax-exps">$\mathit{F}\to\mathscr{F}$</span></a></h2>

<p>&#x5BF9;&#x4E8E;&#x975E;&#x5468;&#x671F;&#x7EDD;&#x5BF9;&#x53EF;&#x79EF;&#x51FD;&#x6570;<span class="mathjax-exps">$f(t)$</span>, &#x5176;<span class="mathjax-exps">$T\to\infty$</span>, &#x6545;<span class="mathjax-exps">$\Omega\to 0$</span>, &#x6B64;&#x65F6;<span class="mathjax-exps">$F_n\to 0$</span>, &#x5373;&#x5176;&#x9891;&#x7387;&#x5206;&#x91CF;&#x5904;&#x7684;&#x5E45;&#x5EA6;&#x8D8B;&#x5411;&#x4E8E;0. &#x6B64;&#x65F6;, <span class="mathjax-exps">$f(t)$</span>&#x5728;&#x67D0;&#x4E2A;&#x65F6;&#x523B;<span class="mathjax-exps">$t$</span>&#x5904;&#x7684;&#x503C;&#x7531;&#x539F;&#x6765;&#x7684;&#x79BB;&#x6563;&#x9891;&#x7387;&#x70B9;&#x7684;&#x65E0;&#x7A77;&#x591A;&#x4E2A;&#x6709;&#x9650;&#x503C;&#x7684;<span class="mathjax-exps">$F_n$</span>&#x53E0;&#x52A0;, &#x53D8;&#x4E3A;&#x8FDE;&#x7EED;&#x9891;&#x7387;&#x70B9;&#x7684;&#x65E0;&#x7A77;&#x591A;&#x4E2A;&#x8D8B;&#x5411;&#x4E8E;0&#x7684;<span class="mathjax-exps">$F_n$</span>&#x53E0;&#x52A0;, &#x4E0D;&#x5229;&#x4E8E;&#x5206;&#x6790;&#x7406;&#x89E3;&#x8BE5;&#x51FD;&#x6570;&#x7684;&#x9891;&#x7387;&#x7279;&#x6027;(&#x5404;&#x9891;&#x7387;&#x70B9;&#x5904;&#x5E45;&#x503C;&#x90FD;&#x8D8B;&#x5411;&#x4E8E;0, &#x770B;&#x4E0D;&#x51FA;&#x4E2A;&#x5565;&#x7279;&#x5F81;&#x51FA;&#x6765;), &#x6211;&#x4EEC;&#x9700;&#x8981;&#x627E;&#x5230;&#x5176;&#x5B83;&#x65B9;&#x6CD5;&#x4EE5;&#x4FBF;&#x4E8E;&#x6211;&#x4EEC;&#x7406;&#x89E3;<span class="mathjax-exps">$f(t)$</span>&#x7684;&#x9891;&#x7387;&#x7279;&#x5F81;.<br>
&#x4ECE;<span class="mathjax-exps">$F_n$</span>&#x7684;&#x516C;&#x5F0F;&#x53EF;&#x4EE5;&#x770B;&#x5230;<span class="mathjax-exps">$F_n$</span>&#x662F;<span class="mathjax-exps">$f$</span>&#x7684;&#x4E00;&#x9636;&#x65E0;&#x7A77;&#x5C0F;&#x91CF;, &#x5373;<span class="mathjax-exps">$\lim_{f\to\infty}\frac{F_n}{f}$</span>&#x662F;&#x4E2A;&#x5E38;&#x6570;, &#x7C7B;&#x6BD4;&#x7269;&#x7406;&#x4E0A;&#x7684;&#x5BC6;&#x5EA6;&#x6982;&#x5FF5;, &#x628A;<span class="mathjax-exps">$\lim_{f\to\infty}\frac{F_n}{f}$</span>&#x8BB0;&#x4E3A;&#x9891;&#x8C31;&#x5BC6;&#x5EA6;. &#x56E0;&#x6B64;, &#x6211;&#x4EEC;&#x4FBF;&#x627E;&#x5230;&#x4E86;&#x4E00;&#x79CD;&#x65B9;&#x6CD5;&#x6765;&#x6765;&#x7EE7;&#x7EED;&#x5206;&#x6790;<span class="mathjax-exps">$f(t)$</span>&#x7684;&#x9891;&#x8C31;&#x7279;&#x5F81;, &#x5373;&#x4E00;&#x7279;&#x5F81;&#x662F;&#x9891;&#x8C31;&#x5728;&#x5404;&#x4E2A;&#x9891;&#x7387;&#x70B9;&#x5904;&#x7684;&#x5BC6;&#x96C6;&#x7A0B;&#x5EA6;, &#x4E5F;&#x5373;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;. <em>&#x5410;&#x69FD;: &#x8FD0;&#x52A8;&#x662F;&#x7EDD;&#x5BF9;&#x7684;, &#x9759;&#x6B62;&#x662F;&#x76F8;&#x5BF9;&#x7684;. &#x4E07;&#x7269;&#x90FD;&#x662F;&#x606F;&#x606F;&#x76F8;&#x5173;&#x7684;, &#x4E00;&#x73AF;&#x6263;&#x4E00;&#x73AF;&#x7684;, &#x6709;&#x65F6;&#x6211;&#x4EEC;&#x4ECE;&#x67D0;&#x4E00;&#x9762;&#x53BB;&#x8BA4;&#x8BC6;&#x67D0;&#x4E2A;&#x8FDC;&#x79BB;&#x6211;&#x4EEC;&#x751F;&#x6D3B;&#x7684;&#x964C;&#x751F;&#x4E8B;/&#x7269;/&#x77E5;&#x8BC6;&#x611F;&#x5230;&#x5403;&#x529B;&#x65F6;, &#x4E5F;&#x8BB8;&#x6362;&#x4E2A;&#x89D2;&#x5EA6;&#x4ECE;&#x53E6;&#x4E00;&#x9762;&#x53BB;&#x770B;&#x5C31;&#x611F;&#x89C9;&#x6BD4;&#x8F83;&#x8F7B;&#x677E;&#x4E86;. &#x4E5F;&#x5373;&#x6211;&#x4EEC;&#x5E38;&#x5E38;&#x5728;&#x5FC3;&#x4E2D;&#x9ED8;&#x5FF5;&#x7684;&quot;&#x6041;&#x548B;&#x5C31;&#x4E0D;&#x80FD;&#x6362;&#x4F4D;&#x601D;&#x8003;&#x4E0B;&#x5450;&quot;!!!</em></p>
<p>&#x6839;&#x636E;&#x4EE5;&#x4E0A;&#x5206;&#x6790;, &#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;&#x63A8;&#x5BFC;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} \mathscr{F}[f(t)] &amp;= F[j\omega] = \lim_{T\to\infty}\frac{F_n}{f}, (n=0,\pm1,\cdots&#x8986;&#x76D6;&#x6240;&#x6709;&#x9891;&#x7387;&#x70B9;) \\ &amp; = \lim_{T\to\infty}\int_{-\infty}^{\infty}f(t)e^{-jn\Omega t} \rm{dt} \\ &amp; = \int_{-\infty}^{\infty}f(t)e^{-j\omega t} \rm{dt} \\ &#x5176;&#x4E2D;n=0,\pm1,\cdots, \lim_{t\to\infty}n\Omega = \omega, &#x5373;&#x79BB;&#x6563;&#x9891;&#x7387;&#x70B9;&#x53D8;&#x4E3A;&#x4E86;&#x8FDE;&#x7EED;&#x9891;&#x7387; \\ \\ f(t) &amp;= \lim_{T\to\infty}\sum_{n=-\infty}^{\infty}F_nTe^{jn\Omega t} \frac{\Omega}{2\pi} \\ &amp; = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)e^{j\omega t}\rm{d\omega} \end{aligned}$$</div><p></p>
<p>&#x7EFC;&#x4E0A;, <span class="mathjax-exps">$f(t)$</span>&#x7684;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;&#x6B63;&#x9006;&#x53D8;&#x6362;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{cases} \mathscr{F}[f(t)] = \int_{-\infty}^{\infty}f(t)e^{-j\omega t} \rm{dt} \\ f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)e^{j\omega t}\rm{d\omega} \end{cases}$$</div><p></p>
<h3 class="mume-header" id="%E5%82%85%E9%87%8C%E5%8F%B6%E7%B3%BB%E6%95%B0%E5%92%8C%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2%E7%9A%84%E5%85%B3%E7%B3%BBtoc"><a href="#toc">&#x5085;&#x91CC;&#x53F6;&#x7CFB;&#x6570;&#x548C;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;&#x7684;&#x5173;&#x7CFB;</a></h3>

<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; F_n = \frac{1}{T}\int_{-\frac{T}{2}}^{\frac{T}{2}}f_0(t)e^{-jn\Omega t}\rm{dt} \\ &amp; F_0(j\omega) = \int_{-\frac{T}{2}}^{\frac{T}{2}}f_0(t)e^{-j\omega t}\rm{dt} \\ \end{aligned}$$</div><p></p>
<p>&#x6839;&#x636E;&#x4E0A;&#x5F0F;&#x6BD4;&#x8F83;&#x53EF;&#x5F97;, <span class="mathjax-exps">$F_n = \frac{1}{T}F_0(j\omega)|_{\omega=n\Omega}$</span>.</p>
<h2 class="mume-header" id="%E4%BB%8E%E5%82%85%E6%B0%8F%E5%8F%98%E6%8D%A2%E5%88%B0%E6%8B%89%E6%B0%8F%E5%8F%98%E6%8D%A2mathscrftomathscrltoc"><a href="#toc">&#x4ECE;&#x5085;&#x6C0F;&#x53D8;&#x6362;&#x5230;&#x62C9;&#x6C0F;&#x53D8;&#x6362;<span class="mathjax-exps">$\mathscr{F}\to\mathscr{L}$</span></a></h2>

<p>&#x6709;&#x4E9B;&#x51FD;&#x6570;&#x4E58;&#x4EE5;<span class="mathjax-exps">$e^{-j\omega t}$</span>&#x5728;&#x5176;&#x5B9A;&#x4E49;&#x57DF;&#x5185;&#x5E76;&#x4E0D;&#x53EF;&#x79EF;(&#x5982;: <span class="mathjax-exps">$e^{\alpha t} \epsilon(t)$</span>), &#x4EA6;&#x5373;&#x8BE5;&#x51FD;&#x6570;&#x7684;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;&#x4E0D;&#x5B58;&#x5728;, &#x4E3A;&#x4E86;&#x89E3;&#x51B3;&#x6B64;&#x7C7B;&#x95EE;&#x9898;, &#x5F15;&#x5165;&#x4E00;&#x4E2A;&#x6307;&#x6570;&#x8870;&#x51CF;&#x56E0;&#x5B50;<span class="mathjax-exps">$e^{-\delta t}$</span>&#x518D;&#x505A;&#x5085;&#x91CC;&#x53F6;&#x53D8;&#x6362;, &#x4FBF;&#x5F97;&#x5230;&#x4E86;&#x62C9;&#x666E;&#x62C9;&#x65AF;&#x53D8;&#x6362;, &#x8FC7;&#x7A0B;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; \mathscr{F}[f(t)e^{-\delta t}] = \int_{-\infty}^{\infty}f(t)e^{-(\delta+j\omega)t}\rm{dt} \\ &amp; f(t)e^{-\delta t} = \frac{1}{2\pi} \int_{-\infty}^{\infty}\mathscr{F}[f(t)e^{-\delta t}]e^{j\omega t} \rm{d\omega} \Rightarrow f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty}\mathscr{F}[f(t)e^{-\delta t}]e^{(\delta + j\omega)t} \rm{d\omega} \end{aligned}$$</div><p></p>
<p>&#x4EE4;<span class="mathjax-exps">$s=\delta + j\omega$</span>, &#x53EF;&#x5F97;&#x62C9;&#x666E;&#x62C9;&#x65AF;&#x53D8;&#x6362;&#x4E3A;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; \mathscr{L}[f(t)] = F(s) = \int_{-\infty}^{\infty}f(t)e^{-st}\rm{dt} \\ &amp; f(t) = \frac{1}{2\pi j} \int_{-\infty}^{\infty} F(s)e^{st}\rm{ds} \end{aligned}$$</div><p></p>
<h2 class="mume-header" id="%E4%BB%8E%E6%8B%89%E6%B0%8F%E5%8F%98%E6%8D%A2%E5%88%B0z%E5%8F%98%E6%8D%A2mathscrltomathscrztoc"><a href="#toc">&#x4ECE;&#x62C9;&#x6C0F;&#x53D8;&#x6362;&#x5230;z&#x53D8;&#x6362;<span class="mathjax-exps">$\mathscr{L}\to\mathscr{z}$</span></a></h2>

<p>&#x8BB0;&#x6709;&#x8FDE;&#x7EED;&#x51FD;&#x6570;<span class="mathjax-exps">$f(t)$</span>, &#x5176;&#x5B58;&#x5728;&#x62C9;&#x6C0F;&#x53D8;&#x6362;<span class="mathjax-exps">$F(s)$</span>, &#x73B0;&#x6BCF;&#x9694;&#x5468;&#x671F;&#x4E3A;T&#x7684;&#x95F4;&#x9694;&#x5BF9;&#x5176;&#x91C7;&#x6837;, &#x8868;&#x8FBE;&#x4E3A;&#x6570;&#x5B66;&#x8FC7;&#x7A0B;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; f_s(t) = \sum_{n=-\infty}^{\infty}f(t)\delta(t-nT) = \sum_{n=-\infty}^{\infty}f(nT)\delta(t-nT) \\ &amp; \mathscr{L}[f_s(t)]  = \int_{-\infty}^{\infty}(\sum_{n=-\infty}^{\infty}f(nT)\delta(t-nT))e^{-st}\rm{dt} \\ &amp; = \sum_{n=-\infty}^{\infty}f(nT)e^{-snT} \end{aligned}$$</div><p></p>
<p>&#x4EE4;<span class="mathjax-exps">$z=e^{sT}$</span>, &#x5F97;&#x5230;<span class="mathjax-exps">$\mathscr{z}$</span>&#x53D8;&#x6362;&#x4E3A;:</p>
<p></p><div class="mathjax-exps">$$\mathscr{f(k)} = \sum_{k=-\infty}^{\infty}f(k)z^{-k}$$</div><p></p>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$z=e^{sT}=\frac{e^{sT/2}}{e^{-sT/2}}\approx\frac{1+sT/2}{1-sT/2}$</span>, &#x8BC1;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; \because s=\frac{1}{T}\ln{z} \\ &amp; \therefore s = \frac{2}{T}[\frac{z-1}{z+1}+\frac{1}{3}(\frac{z-1}{z+1})^3+\frac{1}{5}(\frac{z-1}{z+1})^5 + \cdots] \approx \frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}}) \end{aligned}$$</div><p></p>

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